Tensors are multidimensional arrays which arise in a wide range of applications, including but not limited to topic modeling [1]}, computer version [2]}, collaborative filtering [3]}, and signal processing [4]}. Tensor completion refers to the problem of recovering the target tensor from its partial entries. It is not hard to see that, without any additional assumptions, tensor completion is an ill-posed problem. On the other hand,
this problem can be solved when the target tensor
possesses certain intrinsic low-dimensional structures. A notable example is low-rank tensor completion, where the target tensor is assumed to be low rank. In contrast to the matrix case, tensor has more complex rank notions up to different tensor decompositions such as
CP decomposition [5]}, Tucker decomposition [6]}, tensor train (TT) decomposition [7]} (also known in the computational physics community as matrix product state (MPS) [8]}, [9]}), and hierarchical Tucker (HT) decomposition [10]}.
In this manuscript, we focus on the TT decomposition, a special form of the HT decomposition.
Then the low-rank tensor completion problem can be formulated as follows:
\(\min _{{\cal X}\in \mathbb {R}^{n_1\times n_2\times \cdots \times n_d}}~h\left({\cal X}\right):= \frac{1}{2}\left\Vert  {\cal P}_{\Omega }\left({\cal X}\right)-{\cal P}_{\Omega }\left({\cal T}\right)\right\Vert _{{\footnotesize {\mathsf {F}}}}^2\quad \text{s.t. }~\operatorname{\mathrm {rank}}_{\operatorname{TT}}\left({\cal X}\right)= \mathbf {r},\) 
